The solution of the differential equation `x dy - y dx = sqrt(x^(2) + y^(2)) dx` is

Options

A. `x + sqrt(x^(2) + y^(2)) = cx^(2)`
B. `y - sqrt(x^(2) + y^(2)) = cx`
C. `x - sqrt(x^(2) - y^(2)) = cx`
D. `y + sqrt(x^(2) + y^(2)) = cx^(2)`

Answer

D

Explanation

Given that
`x dy =- ydx = sqrt(x^(2) + y^(2)) dx`
`rArr xdy = (sqrt(x^(2) + y^(2)) + y) dx`
`rArr (dy)/(dx) = (sqrt(x^(2)+y^(2))+y)/(x) `
Now, put `y = vx and (dy)/(dx) = v + x(dv)/(dx)`
`therefore v + x(dv)/(dx)=(sqrt(x^(2) +v^(2)x^(2))+vx)/(x)`
`rArr x (dv)/(dx) = sqrt(1+v^(2)) +v -v = sqrt(1+v^(2))`
On integrating both sides
`int (dv)/(sqrt(1+v^(2))) = int(dx)/(x)`
`log (v + sqrt(1+v^(2))) log x + log c`
`rArr y + sqrt(x^(2) + y^(2)) = cx^(2) " " [because v = (y/(x)]`